... Bill, You might want to look into (modular) exponentiation algorithms, as the calculations x^n (mod p) x^((n-1)/2) (mod p) take almost the sameMessage 1 of 4 , May 5, 2001View SourceOn Sat, 05 May 2001, William F Sindelar wrote:
> Hi Everybody:Bill,
> Fermat says that if A is a PRIME integer, then 2^A divided by A will
> leave a remainder of 2.
> It occurred to me that perhaps we might not have to raise 2 all the way
> to exponent A, but only to some fractional part of A and still have just
> as good a probable prime test as Fermat while significantly reducing
> I came up with 2 statements listed below. They seem to be true but I
> can't prove them. I chased Little Fermat all over the web looking for
> something like this but with no luck. I'm starting to suspect that maybe
> I'm all wet, that I've run into something that works only for small
> numbers. Can anyone help?
> Here are the 2 statements, either of which (if true) could be used as the
> basis of a probable prime test:
> If A is a PRIME integer, then 2^((A+1)/2) divided by A will leave a
> remainder of 2 OR a remainder of (A-2).
> If A is a PRIME integer, then 2^((A-1)/2) divided by A will leave a
> remainder of 1 OR a remainder of (A-1).
> I would appreciate any comments. My thanks and regards to all.
> Bill Sindelar
You might want to look into (modular) exponentiation algorithms, as the calculations
x^n (mod p)
x^((n-1)/2) (mod p)
take almost the same number of steps, in fact only one square and one multiply are required to get from the latter to the former.
A rule of thumb is that an m-bit exponent, n, takes roughly
m squares and m/2 multiplies on average. Here m =~ lg(n) the binary logarithm (round it up).
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Hi there Bill ... the ... What you ve discovered is the basis for two slight improvements on Fermat probable primality. Euler s test does precisely as youMessage 1 of 4 , May 5, 2001View SourceHi there Bill
>Here are the 2 statements, either of which (if true) could be used asthe
>basis of a probable prime test:What you've discovered is the basis for two slight improvements on
>If A is a PRIME integer, then 2^((A+1)/2) divided by A will leave a
>remainder of 2 OR a remainder of (A-2).
>If A is a PRIME integer, then 2^((A-1)/2) divided by A will leave a
>remainder of 1 OR a remainder of (A-1).
Fermat probable primality. Euler's test does precisely as you describe,
and looks at b^((A-1)/2). As you've discovered, it's either +1 or -1
mod A, if A is prime, since it must be a square root of +1. (In fact,
you can determine which result should occur before embarking on the
calculation, and this is sufficient to prove Proth primes, given the
right choice of b).
This is a little less naive than Fermat PRP, and help catch a few
'obvious' pseudoprimes. A little more work will give the Miller-Rabin
test - see http://www.utm.edu/research/primes/prove/prove2_3.html. It's
also the beginnings of the N-1 primality testing methods, where you
consider factors of N-1 other than 2. http://www.utm.edu/research/
primes/prove/prove3_1.html is a gentle introduction to this - notice
the proofs all revolve around "the order of a, modulo p" - the smallest
integer e>0 such that a^e=1 mod p.
As far as execution times of these tests go, the calculation of a^b mod
c can be done in time that scales with the number of bits in b - so
saving one bit in b does not save much. On the other hand, the amount
of extra calculation involved in the theorems on Chris C's page is not
great, so this line of thought can see very strong probable primality
results with not too much effort - and ultimately, deterministic proofs
for those forms where we can factor N-1.